4,819 research outputs found
Multi-latin squares
A multi-latin square of order and index is an array of
multisets, each of cardinality , such that each symbol from a fixed set of
size occurs times in each row and times in each column. A
multi-latin square of index is also referred to as a -latin square. A
-latin square is equivalent to a latin square, so a multi-latin square can
be thought of as a generalization of a latin square.
In this note we show that any partially filled-in -latin square of order
embeds in a -latin square of order , for each , thus
generalizing Evans' Theorem. Exploiting this result, we show that there exist
non-separable -latin squares of order for each . We also show
that for each , there exists some finite value such that for
all , every -latin square of order is separable.
We discuss the connection between -latin squares and related combinatorial
objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares
and -latin trades. We also enumerate and classify -latin squares of small
orders.Comment: Final version as sent to journa
Polyhedral Cones of Magic Cubes and Squares
Using computational algebraic geometry techniques and Hilbert bases of
polyhedral cones we derive explicit formulas and generating functions for the
number of magic squares and magic cubes.Comment: 14 page
A note on higher-dimensional magic matrices
We provide exact and asymptotic formulae for the number of unrestricted,
respectively indecomposable, -dimensional matrices where the sum of all
matrix entries with one coordinate fixed equals 2.Comment: AmS-LaTeX, 9 page
Affine Constellations Without Mutually Unbiased Counterparts
It has been conjectured that a complete set of mutually unbiased bases in a
space of dimension d exists if and only if there is an affine plane of order d.
We introduce affine constellations and compare their existence properties with
those of mutually unbiased constellations, mostly in dimension six. The
observed discrepancies make a deeper relation between the two existence
problems unlikely.Comment: 8 page
An investigation of SUDOKU-inspired non-linear codes with local constraints
Codes with local permutation constraints are described. Belief propagation decoding is shown to require the computation of permanents, and trellis-based methods for computing the permanents are introduced. New insights into the asymptotic performance of such codes are presented. A universal encoder for codes with local constraints is introduced, and simulation results for two code structures, SUDOKU and semi-pandiagonal Latin squares, are presented.This is the author accepted manuscript. The final version is available from IEEE via 10.1109/ISIT.2015.728279
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
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